TY - JOUR
T1 - Asymptotic behavior of the estimation error covariance of quaternion estimators
AU - Carmi, Avishy
AU - Oshman, Yaakov
N1 - Funding Information:
This research was supported by the Israel Science Foundation under grant no. 1032/04 and by the Technion–Israel Institute of Technology’s Asher Space Research Fund. The authors thank John L. Junkins for his comment that motivated Sec. IV and F. Landis Markley for helpful discussions.
PY - 2008/1/1
Y1 - 2008/1/1
N2 - The asymptotic behavior of the estimation error covariance of quaternion estimators is mathematically examined. It is proved that the condition number of the asymptotic covariance matrix is of the order of the inverse of its largest eigenvalue, so that this matrix becomes asymptotically ill-conditioned as its trace tends to zero. Nevertheless, it is proved that the aforementioned asymptotic behavior cannot be captured by low-order Taylor approximations of the covariance, such as the one computed by the extended Kalman filter. Geometrical interpretation of the results is provided, using tools from differential geometry. The analytical results are demonstrated via a simulation study using the recently introduced quaternion particle filter and the additive quaternion extended Kalman filter.
AB - The asymptotic behavior of the estimation error covariance of quaternion estimators is mathematically examined. It is proved that the condition number of the asymptotic covariance matrix is of the order of the inverse of its largest eigenvalue, so that this matrix becomes asymptotically ill-conditioned as its trace tends to zero. Nevertheless, it is proved that the aforementioned asymptotic behavior cannot be captured by low-order Taylor approximations of the covariance, such as the one computed by the extended Kalman filter. Geometrical interpretation of the results is provided, using tools from differential geometry. The analytical results are demonstrated via a simulation study using the recently introduced quaternion particle filter and the additive quaternion extended Kalman filter.
UR - http://www.scopus.com/inward/record.url?scp=57149131195&partnerID=8YFLogxK
U2 - 10.2514/1.35781
DO - 10.2514/1.35781
M3 - Article
AN - SCOPUS:57149131195
SN - 0731-5090
VL - 31
SP - 1665
EP - 1676
JO - Journal of Guidance, Control, and Dynamics
JF - Journal of Guidance, Control, and Dynamics
IS - 6
ER -